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Building more math success in Grades 7 – 10
Marian SmallApril 2013
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Focusing math instruction on what students need to know, not just what they can do
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The outcomes
• , as they are presented in the curriculum document, speak more to what students need to do than what they need to know.
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For example, Grade 7
• Solve percent problems involving percents from 1% to 100%.
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So what do you think matters most?
Poll:• A: Recognizing the three different types
of percent problems and realizing what’s different about solving them
• B: Being able to estimate the answer to a percent problem.
• C: Recognizing the equivalent fractions and decimals for a percent.
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What do you think of this?
• That students realize that every percent problem involves renaming a ratio in the form []/100 in another way.
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For example…
• Why is finding the sale price actually renaming 40/100 as []/70?
$70
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For example…
• I paid $30 on sale.• What was the original price?
• Why is renaming the original price really renaming 40/100 as 30/[]?
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For example…
• I paid $30 instead of $80.• What was the percent paid?
• Why is finding the percent paid really renaming 30/80 as []/100?
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Another important idea…
• That students realize that knowing any percent of a number tells you about any other percent.
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For example…
• I tell you that 20% of a certain number is 42.
• What other percents of that number do you know even before you figure out the number?
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Write the percents you know
• on the blank screen.
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If I tell you…
• 15% of a number, how could you figure out 75% of that number?
• Raise your hand to respond.
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If I tell you…
• 15% of a number, how could you figure 5% of that number?• Raise your hand to respond.
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If I tell you…
• 15% of a number, how could you figure out 50% of that number?• Raise your hand to respond.
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Another important idea
• That renaming a percent as a fraction or decimal sometimes helps solve problems.
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For example…
• I want to figure out how much I can withdraw from my bank account if Mom says 25% maximum.
• I have $424.• How could I figure out the amount in my
head?
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Which of these problems also makes the point?
Poll• A: estimating 35% of 612• B: calculating 10% of 417• C: calculating 43% of 812• D: two of the above
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Let’s look at Grade 8
• Model and solve problems concretely, pictorially and symbolically, using linear equations of the form:
• ax=b• x/a = b,a=0 • ax+b=c• x/a+b=c,a=0• a(x+b)=c where a, b and c are integers.
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What matters
• besides just solving?
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I think that most important is…
POLL:• A: Using more than one strategy.• B: solving symbolically• C: Estimating solutions• D: Checking that a solution is correct by
substituting
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Maybe estimation
• We want students to come up with a reasonable estimate for a solution without solving first.
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So I might ask…
• Is the solution to 3x/4 – 12 = 6 closer to 0, 10, or 20? How do you know?
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So I might ask…
OR• The solution to an equation is close to
40, but not exactly 40. What might the equation be?
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So I might ask…
OR• How might I estimate the solution to
5x – 80 = 300 without actually solving it?
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So I might ask…
OR• Can you estimate the solution to the
equation 3x + [] = 90 by ignoring the [] and just saying 30?
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Another important idea
• That the same equation could represent very different problems.
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So I might ask…
• Write a real-life problem that might be solved using the equation x/4 – 12 = 10.
• How are our problems alike? Different?• But let’s start with something just a tad
simpler since we’re online.
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Which problem relates to x/4 – 12 = 10?
POLL• A: There were 4 kids sharing a prize. They gave
$12 away and there was $10 left. What was the amount of the prize?
• B: 4 kids shared a prize. One kid gave $12 away and still had $10 left. What was the amount of the prize?
• C: 4 kids shared a prize. One kid gave $12 away and still had $10 left. What was each kid’s share?
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Or…
• Represent each problem on the next slide with an equation. What do you notice? Why does that make sense?
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Or…
• Problem 1: The perimeter of a regular hexagon is 90 cm. What is each side length?
• Problem 2: A rectangle’s length is twice its width. The total perimeter is 90 cm. What is the width?
• Write your response on the whiteboard or raise your hand.
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Problem 1: The perimeter of a regular hexagon is 90 cm. What is each side length?
Problem 2: A rectangle’s length is twice its width. The total perimeter is 90 cm. What is the width?
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And the flip side…
• You can always represent a problem with an equation in more than one way.
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For example…
• Write an equation to represent this problem : Jennifer had twice as many apps as Lia. Together, they had 78 apps. How many did each have?
• What other equation could you have written instead?
• Write one equation on the whiteboard.
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Or…
• A problem is represented by the equation 2x + 18 = 54.
• What could the problem have been?• What other equation could have
represented the problem?• Which equation do you like better? Why?
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Another important point
• That modelling an equation to help you solve it always involves some sort of “balance”.
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For example….
• Where are the 3, the multiplication inside of 3x, the x, the 5, the 26 and the = in the picture below that represents the equation 3x + 5 = 26?
• You raise your hand about where you see the 3 and the multiplication.
X X X 5 26
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Or…
• Some people say that an equation (e.g. 4x – 5 = 19) describes a balance. What do they mean?
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Or…
• How could you use a pan balance to model the equation 3x + 8 = 29?
• Why does the model make sense?
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Another important point…
• Solving an equation means writing an equivalent equation that is easier to interpret. (e.g. We rewrite 3x – 8 = 19 as 3x = 27 or x = 9 since they say the same thing but it’s quicker to see what x is.)
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So we might ask…
• Why might it be useful to rewrite the equation 4x + 18 = 66 as 4x = 48 in order to solve it?
• Why are you allowed to do that?
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Or we could ask…
• Why might someone call the equations 3x – 5 = 52 and 3x = 57 equivalent?
• Which would you rather solve? Why?
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Do we have
• Grade 10 teachers on line?• If so, we will continue with the
presentation.• If not, we will engage in a conversation
about how to do this work with other expectations.
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Grade 10
• Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.
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Of course…
• Students need to learn the definitions of sine, cosine, tangent. These matter, but they are not ideas.
• But what ideas do they need to learn?
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What do you think is important?
• POLL:• A: to predict whether sine, cosine or
tangent is greater for particular angles• B: to learn that sin2 + cos2 = 1• C: to learn that the size of sine or cosine
is independent of triangle size
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Maybe…
• The size of the trig function has nothing to do with the size of the triangle, i.e. a big triangle and little triangle can have the same sine, cosine and/or tangent.
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So you might ask…
• The sine of an angle in a right triangle is 0.42.
• Is it more likely that the hypotenuse is 1 cm, 10 cm or 100 cm, or don’t you know? Explain.
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You might want them to know..
• That a bigger angle (in a 90° triangle) has a bigger sine and a smaller cosine and why, but that the change in angle size is not proportional to the change in sine or cosine.
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So you might ask…
• The sine of <A is 0.2 greater than the sine of <B.
• Do you know which angle is greater? Explain.
• Do you know how much greater the bigger angle is? Explain.
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You might want students to realize…
• That even though sines and cosines have to stay 1 or less, tangents can get really big and why.
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So you might ask….
Which statements below are true? Explain.
• The sine of an angle can never be 2.• The tangent of a small angle can be 2.• The tangent of a large angle can be 5.• There is no greatest possible tangent.
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You might want students to…
• Have a sense of trig ratio relationships, e.g. when sine > cosine, that tan > sine, etc.
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So you might ask… Consider each statement. The angles are all less than
90°. Is the statement always, sometimes or never true? Use the pen tool to write a check or x to indicate your thoughts.
• sin A > sin B when A < B• cos A > cos B when A < B• tan A > sin A• cos A < tan A• sin A = cos B
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To conclude
• The work we are talking about involves looking deeply at outcomes to focus on the ideas that are critical to really understanding what is going on.
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To conclude
• It is not about the complexity of questions students can answer.
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Download
• Download these slides at •www.onetwoinfinity.ca(Alberta7-10 webinar)• ERLC wiki at http://goo.gl/LxO95